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Suppose that $\{\lambda_n\}_{n=1}^{\infty}$ is a sequence of distinctpositive real numbers satisfying the conditions inf$\{\lambda_{n+1}-\lambda_n\}>0,$ and $\sum_{n=1}^{\infty}\lambda_n^{-1}<\infty.$ We prove that theexponential system $\{e^{\lambda_n t}\}_{n=1}^{\infty}$ is hereditarilycomplete in the closure of the subspace spanned by $\{e^{\lambda_nt}\}_{n=1}^{\infty}$ in the space $L^2 (a,b)$. We also give an example of aclass of compact non-normal operators defined on this closure which admitspectral synthesis.

Hereditary completeness of Exponential systems $\{e^{\lambda_n t}\}_{n=1}^{\infty}$ in their closed span in $L^2 (a, b)$ and Spectral Synthesis